DEPENDENCE OF EIGENVALUES OF STURM-LIOUVILLE PROBLEMS ON THE BOUNDARY

The eignvalues of Sturm-Liouville (SL) problems depend not only contin uously but smoothly on boundary points. The derivative of the nth eige nvalue as a function of an endpoint satisfies a first order differenti al equation. This for arbitrary (separated or coupled) self-adjoint re gular boundary conditions. In addition, as the length of the interval shrinks to zero all higher eignvalues march off to plus infinity. This is also true for the first (i.e., lowest) Dirichlet eigenvalue but no t for the lowest Neumann eigenvalue. The latter has a finite limit. (C ) 1996 Academic Press, Inc.